Relative fragmentation in ternary systems within the temperature-dependent relativistic mean-field approach

For the first time, we apply the temperature-dependent relativistic mean-field (TRMF) model to study the ternary fragmentation of heavy nuclei using the level density approach. The relative fragmentation probability of a particular fragment is obtained by evaluating the convolution integrals that employ the excitation energy and the level density parameter for a given temperature calculated within the TRMF formalism. To illustrate, we have considered the ternary fragmentations in ${}^{252}\mathrm{Cf}, {}^{242}\mathrm{Pu}$, and ${}^{236}\mathrm{U}$ with a fixed third fragment $A_3={}^{48}\mathrm{Ca}, {}^{20}\mathrm{O}$, and ${}^{16}\mathrm{O}$, respectively. The relative fragmentation probabilities are studied for the temperatures $T=1$, 2, and 3 MeV. For the comparison, the relative fragmentation probabilities are also calculated from the single-particle energies of the finite range droplet model (FRDM). In general, the larger phase space for the ternary fragmentation is observed indicating that such fragmentations are most probable ones. For $T=2$ and 3 MeV, $\mathrm{Sn}+\mathrm{Ni}+\mathrm{Ca}$ is the most probable combination for the nucleus ${}^{252}\mathrm{Cf}$. However, for the nuclei ${}^{242}\mathrm{Pu}$ and ${}^{236}\mathrm{U}$, the maximum fragmentation probabilities at $T=2$ MeV differ from those at $T=3$ MeV. For $T=3$ MeV, the closed shell ($Z=8$) light-mass fragment with its corresponding partners has larger scission point probabilities. But, at $T=2$ MeV, Si, P, and S are favorable fragments with the corresponding partners. It is noticed that the symmetric binary fragmentation along with the fixed third fragment for ${}^{242}\mathrm{Pu}$ and ${}^{236}\mathrm{U}$ is also favored at $T=1$ MeV.

Figure 1

Ternary fragmentation of ${}^{252}\mathrm{Cf}$ for the fixed third fragment ${}^{48}\mathrm{Ca}$ for the temperatures $T=1$, 2, and 3 MeV. The total fragmentation probabilities are normalized to 2.

Figure 2

Ternary fragmentation of ${}^{242}\mathrm{Pu}$ for the fixed third fragment ${}^{20}\mathrm{O}$ for the temperatures $T=1$, 2, and 3 MeV. The total fragmentation probabilities are normalized to 2.

Figure 3

Ternary fragmentation of ${}^{236}\mathrm{U}$ for the fixed third fragment ${}^{16}\mathrm{O}$ for the temperatures $T=1$, 2, and 3 MeV. The total fragmentation probabilities are normalized to 2.

Figure 4

The level density parameter $a$ of the ternary fragmentation of ${}^{252}\mathrm{Cf}$ for the temperatures $T=1$, 2, and 3 MeV within the TRMF and FRDM formalisms.

Figure 5

The level density parameter $a$ of the ternary fragmentation of ${}^{242}\mathrm{Pu}$ for the temperatures $T=1$, 2, and 3 MeV within the TRMF and FRDM formalisms.

Figure 6

The level density parameter $a$ of the ternary fragmentation of ${}^{236}\mathrm{U}$ for the temperatures $T=1$, 2, and 3 MeV within the TRMF and FRDM formalisms.

Figure 7

The level density of the ternary fragmentation of ${}^{252}\mathrm{Cf}$ for the temperatures $T=1$, 2, and 3 MeV.

Figure 8

The level density of the ternary fragmentation of ${}^{242}\mathrm{Pu}$ for the temperatures $T=1$, 2, and 3 MeV.

Figure 9

The level density of the ternary fragmentation of ${}^{236}\mathrm{U}$ for the temperatures $T=1$, 2, and 3 MeV.